The fact that the Reidemeister moves of type 3 are not true for (the algebraic version of) the emergent algebras, i.e. that the translations of those cannot be expressed as a sequence of moves from graphic lambda calculus, is a feature of the formalism and not a weakness. This is explained in detail in sections 5, 6 [1]}, but unfortunately at the moment of the writing that article the graphic lambda calculus was not available. It is an interesting goal the one of expressing the constructions from the mentioned sections as statements about the computability in the sense of definition REF  of the translations of certain tangle diagrams.
